list or set of polynomially nonlinear PDEs or ODEs (may contain inequations)

vars

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(optional) list of the main dependent variables

options

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(optional) sequence of options to control the behavior of rifsimp

Description

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The rifsimp function can be used to simplify or rework overdetermined systems of polynomially nonlinear PDEs or ODEs and inequations to a more useful form -- relative standard form. The rifsimp function does not solve PDE systems, but provides existence and uniqueness information, and can be used as a first step to their solution. As an example, inconsistent systems can be detected by rifsimp.

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Basically, given an input PDE system, and a list of dependent variables or constants to solve for, rifsimp returns the simplified PDE system along with any existence conditions required for the simplified system to hold.

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Detailed examples of the use of rifsimp for various systems (along with some explanation of the algorithm) can be found in rifsimp[overview].

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Other options are sometimes required along with the specification of the system and its solving variables. For common options, please see rifsimp[options], and for more advanced use, please see rifsimp[adv_options].

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For a description of all possible output configurations, see rifsimp[output].

Examples

1. Overdetermined Systems

As a first example, we have the overdetermined system of two equations in one dependent variable y(x).

In this case, it gives us the expected results for a second order ODE, but it can also calculate the required initial data for more complex PDE systems (see initialdata for more information). Numerical methods can now be successfully applied to the reduced system, with initial conditions of the type calculated by initialdata.

This example shows the use of rifsimp as a preprocessor for a constrained mechanical system (that is, a Differential-Algebraic Equation or DAE system). The method of Lagrange formulates the motion of a bead of nonzero mass m on a frictionless wire of shape Phi(x,y) under gravity as follows:

Such constrained systems present great difficulties for numerical solvers. Of course, we could eliminate the constraint using polar coordinates, but in general this is impossible for mechanical systems with complicated constraints. For example, just replace the constraint with another function, so that the wire has a different shape.

>

&Phi;2 ≔ x&comma;y&rarr;x4&plus;y4−1

&Phi;2 ≔ x&comma;y&rarr;x4&plus;y4−1

(13)

(See Reid et al., 1996, given in the reference and package information page Rif). The pendulum is the classic example of such systems that have been the focus of much recent research due to their importance in applications.

Although there is no simplification as in the previous examples, rifsimp has found additional constraints that the initial conditions must satisfy (Constraint and Pivots in the initialdata output), and obtained an equation for the time derivative of lambda(t). Maple's dsolve[numeric] can then be used to obtain a numerical solution, in the manner of the previous example.

4. Lie Symmetry Determination

This example shows the use of rifsimp to assist in determination of Lie point symmetries of an ODE. Given the ODE:

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